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An Extension of Stochastic Green’s Function Method to Long-Period Strong
Ground-motion Simulation
Y. Hisada and J. Bielak
Purpose: Extension of Stochastic Green’s Function to Longer Periods
Realistic Phases
・ Random Phases at Shorter Period
・ Coherent Phases at Longer Periods
→ Directivity Pulses, Fling Step, Seismic Moment Realistic Green’s Functions (e.g., surface wave)
・ Green’s Functions of Layered Half-Space
→ Easy to compute them at shorter periods
(e.g., Hisada, 1993, 1995)
Broadband Strong Ground Motion Simulation (Hybrid Methods)
Short period ( < 1 s ): Stochastic and empirical methods ( ex., Stochastic Green’s function method ) → omega-squared model, random phases
Long period ( > 1 s ): Deterministic methods ( FDM, FEM, Green’s functions for layered media ) → coherent phases (e.g., directivity pulses), seismic moment
0 1 2 period (s)
short ←→ long period
M7 Eq.
0 1 2 4 period (s)
M8 Eq.
short ←→ long period
Broadband Strong Ground Motion Simulation (Hybrid Methods)
The crossing period is around 1 sec. Ok for M7 eq., but not for M8 eq. → Resolution for M8 eq. is not fine enough at 1 sec (e.g., Size of sub-faults is 10 – 20 km.) Extension of deterministic methods to shorter periods.
0 1 2 period (s)
short ←→ long period
M7 Eq.
0 1 2 4 period (s)
M8 Eq.
short ←→ long period
Modified K-2 model ( Hisada, 2000 )
0
0.5
1
1.5
2
2.5
3
0 0.5 1 1.5 2
time (sec)
Slip
Vel
ocity
k2 slip distribution k2 rupture time Kostrov-type slip velocitywith fmax
・ Slip and rupture time are continuous on a fault plane・ Large number of source points at shorter periods・ Ok for FEM (FDM), but not for theoretical methods using Green’s Function of Layered half-space
1/k2
0 k (wavenumber)
ampl
itude
Phase: coherent random
Source spec.: ω2 model
1.0E-04
1.0E-03
1.0E-02
1.0E-01
1.0E+00
0.01 0.1 1 10 100
frequency (Hz)
Sour
ce S
pect
ra 1/ω2
frequencyk2 model
Broadband Strong Ground Motion Simulation (Hybrid Methods)
Short period ( < 1 s ): Stochastic and empirical methods ( ex., Stochastic Green’s function method ) → omega-squared model, random phases
Long period ( > 1 s ): Deterministic methods ( FDM, FEM, Green’s functions for layered media ) → coherent phases (e.g., directivity pulses), seismic moment
0 1 2 period (s)
short ←→ long period
M7 Eq.
0 1 2 4 period (s)
M8 Eq.
short ←→ long period
Stochastic Green’s Function Method (Kamae et al., 1998) : Boore’s Source Model + Irikura’s Empirical Green’s Function Summation Method
→ Fast Computation: One source point per sub-fault→ Green’s Functions of the Far-Field S Wave (1/r)
Observation Point
Seismic Fault
Moment Rate Function with ω2 Model (Ohnishi and Horike, 2000)
Far-Field S-waves from a point source
Far-Field S-waves from Boore’s source model
Moment Rate Function for ω2 model
)()()(
)( max00
0 fPfSMdt
MdMi c
Slip Velocity for ω2 model
Representation Theorem for ω2 model
For Point Dislocation Source
)()()(
)( maxfPfDSdt
DdDi c
deUneneDU rtijikijjik
)())(()( *,
)())(()( *,0 jikijjik UneneMU
Boore’s Source Model with Random Phases
ω2 Amplitude
+ Random Phases
fc=1 Hz
fmax=10 Hz
Moment Rate Function
( Slip Velocity Function )FIT withTime Window
・ Unstable and Incoherent at Longer Periods→ ○Acceleration×Directivity Pulses ×Fling Step×Seismic Moment
Example 1
Example 2
Boore’s Source Model with Zero Phases (Coherent Phases)
ω2 Amplitude + Zero Phases
fc=1 Hz
fmax=10 Hz
Moment Rate Function
( Slip Velocity Function )
-0.5
0
0.5
1
1.5
0 2 4 6 8 10
time (sec)
slip
vel
octy
Moment Rate +1/fc sec delay
Moment Function
・ Smoothed Ramp Function→ ×Acceleration ○Directivity Pulses○Fling Step ○Seismic Moment
FIT withNo Time Window
Boore’s Source Model with Zero and Random Phases (Introduction of fr) ω2 Amp. + Zero and
Random Phases
fc=1 Hz
fmax=10 Hz
Moment Rate Function
( Slip Velocity Function )
fr=1 HzMoment Function
Moment Rate +1/fc sec delay
・ Ramp Function with high freq. ripples→ ○Acceleration ○Directivity Pulses ○Fling Step○Seismic Moment
FIT withTime Window
Example (Boore’s Source with Zero Phases) : r=20 km, Vp=5,Vs=3km/s
R=20 km
R=20 km
45° S wave
P wave
Moment Rate Function1. Triangle ( τ=1s )2. ω 2 Model ( fc=1 Hz fmax=10 Hz, 0 phases )
Point Source at 20 km
0.00001
0.0001
0.001
0.01
0.1
1
10
0.01 0.1 1 10 100
frequency (Hz)
Fo
uri
er
Dis
pla
ce
me
nt
Am
plit
ud
e
(cm
*se
c)
x11(20,0)-no rand
x11(14.1,14.1)-1/r & no rand
x11(20,0)-grfltHF1
x11(14.1,14.1)-grfltHF2
Far Field Terms at r = 20 km (Point Source)
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10 11 12
time (sec)
dis
pla
ce
me
nt
(X)
Current Model (S wave)
Current Model (P wave)
Triangle Slip Model (S wave)
Triangle Slip Model (P wave)
Far Field Displacement
P Wave
S Wave
S Wave
P Wave
Boore’s Source with Zero and Random Phases
acc
-400
-300
-200
-100
0
100
200
300
400
0 1 2 3 4 5 6 7 8 9 10 11 12
acc
point source and far field term at (20,0) km
-2
-1
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9 10 11 12
time (sec)
dis
p
x11(20,0)-1/r & idum=1 &fran=1 HzFar-Field
Displacement
S Wave
0.0001
0.001
0.01
0.1
1
10
0.01 0.1 1 10 100
frequency (Hz)
Mo
men
t R
ate
original
result
Far-Field Acceleration
Proposed Modelfc=1 Hzfmax=10 Hzfr=1 H z
S Wave
Summarized Results (Three Models)
- 3
2
7
12
17
22
27
0 10 20time (s)
Dis
plac
emen
t (c
m)
TriangleSlipVelocity
Boore'sAmplitude+ 0 Phases
Boore'sAmplitude+ fr=1 Hz
- 40
- 20
0
20
40
60
80
100
120
140
0 10 20
time (s)
Vel
ocity
(cm
/s)
-600
-100
400
900
1400
1900
2400
0 10 20
time (s)
acce
lera
tion
(gal
) Displacement Velocity Acceleration
Triangle Slip Velocity
ω2 Model + 0 Phases
ω2 Model + 0 & Random Phases
Summary
We extended a stochastic Green’s function method to longer periods in order to simulate coherent waves, by introducing zero phases at frequencies smaller than fr (a corner frequency).
We can easily incorporate this method with more realistic Green’s functions, such as those of layered half-spaces.